adding and subtracting negative numbers problem solving

Negative Number Addition-Subtraction Quiz

How to Add and Subtract Positive and Negative Numbers

Numbers can be positive or negative.

This is the Number Line :

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Example: 5 is really +5

Play with it!

On the Number Line positive goes to the right and negative to the left.

Try the sliders below and see what happens:

Balloons and Weights

Let us think about numbers as balloons (positive) and weights (negative):

This basket has balloons and weights tied to it:

• The balloons pull up ( positive )
• And the weights drag down ( negative )

Adding a Positive Number

Adding positive numbers is just simple addition.

We can add balloons (we are adding positive value)

the basket gets pulled upwards (positive)

Example: 2 + 3 = 5

is really saying

"Positive 2 plus Positive 3 equals Positive 5"

We could write it as (+2) + (+3) = (+5)

Subtracting A Positive Number

Subtracting positive numbers is just simple subtraction.

We can take away balloons (we are subtracting positive value)

the basket gets pulled downwards (negative)

Example: 6 − 3 = 3

"Positive 6 minus Positive 3 equals Positive 3"

We could write it as (+6) − (+3) = (+3)

Adding A Negative Number

Now let's see what adding and subtracting negative numbers looks like:

We can add weights (we are adding negative values)

Example: 6 + (−3) = 3

"Positive 6 plus Negative 3 equals Positive 3"

We could write it as (+6) + (−3) = (+3)

The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.

So these have the same result :

• (+6) − (+3) = (+3)
• (+6) + (−3) = (+3)

In other words subtracting a positive is the same as adding a negative .

Subtracting A Negative Number

Lastly, we can take away weights (we are subtracting negative values)

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Yes indeed! Subtracting a Negative is the same as adding!

Two Negatives make a Positive

What Did We Find?

Adding a positive number is simple addition ..., positive and negative together ..., example: what is 6 − (+3) .

6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7) ?

5+(−7) = 5 − 7 = −2

Subtracting a negative ...

Example: what is 14 − (−4) .

14−(−4) = 14 + 4 = 18

It can all be put into two rules :

They are "like signs" when they are like each other (in other words: the same).

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign

Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign .

5+(−2) = 5 − 2 = 3

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign .

25−(−4) = 25+4 = 29

Starting Negative

What if we start with a negative number?

Using The Number Line can help:

Example: What is −3+(+2) ?

+(+) are like signs, so they become a positive sign .

−3+(+2) = −3 + 2

−3+(+2) = −3 + 2 = −1

Example: What is −3+(−2) ?

+(−) are unlike signs, so they become a negative sign .

−3+(−2) = −3 − 2

−3+(−2) = −3 − 2 = −5

Now Play With It!

A Common Sense Explanation

And there is a "common sense" explanation:

If I say "Eat!" I am encouraging you to eat (positive)

If I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you are done!

Another Common Sense Explanation

A friend is + , an enemy is −

A Bank Example

Example: last year the bank subtracted $10 from your account by mistake, and they want to fix it..

So the bank must take away a negative $10 .

Let's say your current balance is $80, so you will then have:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

A Long Example You Might Like

Ally's points.

Ally can be naughty or nice. So Ally's parents have said

"If you are nice we will add 3 points (+3). If you are naughty, we take away 3 points (−3). When you reach 30 Points you get a toy."

So when we subtract a negative, we gain points (i.e. the same as adding points).

See: both " 15 − (+3) " and " 15 + (−3) " result in 12.

It doesn't matter if you subtract positive points or add negative points, you still end up losing points.

Try These Exercises ...

Now try This Worksheet , and see how you go.

And also try these questions:

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Addition of positive integers including regrouping

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Add & subtract neg. nos.

Adding and subtracting negative numbers

Here is everything you need to know about adding and subtracting negative numbers including adding and subtracting positive and negative numbers.

Students will first learn about adding and subtracting negative numbers as part of the number system in 7 th grade.

What is adding and subtracting negative numbers?

Adding and subtracting negative numbers is operating with non-positive numbers. Negative integers are numbers that have a value less than zero, like - \; 5 and - \; 21.

To do this, you can use the number line. If you are adding, move to the right of the number line. If you are subtracting, move the opposite direction, to the left, of the number line.

For example,

To solve - \; 5+6 on a number line, start at - \; 5 and move 6 to the right.

To solve - \; 1-3, start at - \; 1 on the number line and move left 3.

- \; 1-3=- \; 4

Using the number line in this way can help you solve each type of problem, except when adding or subtracting a negative number . For this, a step needs to be added.

There are a few ways to think about how to solve this:

• Think about the - symbol as the opposite \rightarrow so - \; ( - \; 2) means “the opposite of - \; 2 ” which is + \; 2.
• Consider negatives to be cold and positives to be warm. That means - \; ( - \; 2) is like taking away cold. When you take away cold, it gets warmer. So 2 warmer than - \; 5.

All three ways show that - \; 5-( - \; 2)=- \; 3. From this you can establish the rule that subtracting a negative is the same as adding a positive.

The grid below can help to work out whether to add or subtract with a positive or negative number.

If you have two signs next to each other, change them to a single sign.

• If the signs are the same, add.
• If the signs are different, subtract.

Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – The Numbers System (7.NS.A.1d) Apply properties of operations as strategies to add and subtract rational numbers.

[FREE] Addition and Subtraction Worksheet (Grade 2, 3, 4 and 7)

Use this quiz to check your grade 2, 3, 4 and 7 students’ understanding of addition and subtraction. 15+ questions with answers covering a range of 2nd, 3rd, 4th and 7th grade addition and subtraction topics to identify areas of strength and support!

How to add and subtract negative numbers

In order to add and subtract negative numbers:

Change two adjacent signs to a single sign.

Circle the first number on the number line.

Use the number line to add or subtract.

Write your final answer.

Adding and subtracting negative numbers examples

Example 1: adding a positive number.

Solve - \; 4+7.

In this case you do not have two signs next to each other so no signs change.

2 Circle the first number on the number line.

The first number in the question is - \; 4.

3 Use the number line to add or subtract.

As you are calculating + \; 7, move 7 spaces right from - \; 4 on the number line:

4 Write your final answer.

Example 2: adding a negative number

Solve - \; 2+- \; 3.

In this case you have the two symbols +- next to each other. As the signs are different, subtract 3 from - \; 2.

- \; 2+- \; 3=- \; 2-3

The first number in the question is - \; 2.

In this case, you are subtracting 3 , so move 3 spaces left from - \; 2 on the number line:

Example 3: subtracting a positive number

Solve - \; 5-2.

In this case, you do not have two signs next to each other, so no signs change.

The first number in the question is - \; 5.

In this case, you are calculating - \; 2 so move 2 spaces left from - \; 5 on the number line:

Example 4: subtracting a negative number

Solve: - \; 8-- \; 10.

In this case, you have a minus and a minus next to each other (--).

Since the signs are the same, replace them with an addition.

- \; 8-- \; 10=- \; 8+10

The first number in the question is - \; 8.

In this case you are calculating + \; 10, so move 10 spaces right from - \; 8 on the number line:

Example 5: mixed operations

Answer the calculation below:

The - \; 5 has been placed in brackets to highlight that the number is negative. Removing or inserting the brackets here does not change the calculation. You can therefore write the same calculation as 7-8-- \; 5.

Since the signs are the same, these two signs are replaced with an addition.

7-8-- \; 5=7-8+5

Note: no other signs change.

The first number in the question is 7.

In this case, you are calculating - \; 8 , so move 8 spaces left from 7 on the number line:

Nex,t you are calculating + \; 5 , so move 5 spaces right from - \; 1 on the number line:

Example 6: word problem

Alina had \$ 12 in her bank account. She purchased a t-shirt for \$ 20. By how much did she overdraw her account?

To answer this question, first write an equation that expresses the question.

In this case, you do not have two signs next to each other so no signs need to change.

The first number in the question is 12.

In this case you are calculating - \; 20.

Instead of subtracting 1 each time, each jump is subtracting 2 so we can subtract 20 in 10 jumps.

She is overdrawn by \$ 8.

Teaching tips for adding and subtracting negative numbers

• Use visuals, such as number lines or counters, to illustrate the concept of adding and subtracting negative numbers.
• When teaching rules, such as when you add two negative numbers you get a positive, make sure to provide an explanation why it works and provide a visual if needed.
• When providing students with practice worksheets, make sure to begin with simple problems and progress to more challenging problems as students become confident with the concept.
• For students that are struggling with mastering the concept, consider the use of student math tutors. This is a strategy where students are in charge of the tutorials for other students. Sometimes allowing other students to use student language allows for deeper understanding.

Easy mistakes to make

• Greater negative does mean a larger number Students sometimes assume that the larger a negative number the greater it is. For example, students might incorrectly assume - \; 3 is greater than 2 because 3 is a larger number.
• Using the rules for two signs when the signs are not together For example, thinking that - \; 5+7 would change to 5-7 (since there is a \; + and a \; - ). Changing the signs only applies when the signs are together. So in the case of - \; 5+7, nothing would change and you would start at - \; 5 on the number line and move 7 spaces to the right. If the calculation was 5-+7 and the signs were together, you would then change it to 5-7.

Related addition and subtraction lessons

• Adding and subtracting integers
• Adding and subtracting rational numbers
• Add and subtract within 100
• Standard algorithm addition
• Order of operations
• Absolute value

Practice adding and subtracting negative numbers questions

1. Solve: – \; 6 + 10

Start at – \; 6 on the number line and move to the right 10 numbers.

2. Solve: – \; 4 +- \; 8.

There is a positive (+) and negative sign (-) together so change them to a subtraction (-) only.

Start at – \; 4 on the number line and move to the left 8 numbers. As each jump is – \; 2, this is 4 jumps.

3. Solve: – \; 9-12.

Start at – \; 9 on the number line and move to the left 12 numbers.

4. Solve: – \; 15- ( – \; 6).

There are two negatives (-) together, so change the signs to an addition (+) sign only.

Start at – \; 15 on the number line and move to the right 6 numbers.

5. Solve: – \; 7+ ( – \; 8)-5+2.

There is a positive (+) and a negative (-) together, so change the signs to a subtraction (-) sign only.

Start at – \; 7 and move 8 places to the left, then move another 5 places to the left, then move 2 places to the right.

6. At 5 am in the morning, the temperature in Fargo, North Dakota was – \; 9^{\circ} \mathrm{F}. By 10 am the temperature had risen by 11^{\circ} \mathrm{F}. What was the new temperature?

The equation you need to solve is – \; 9+11.

Start at – \; 9 on the number line and because it is an addition problem, move to the right 11 numbers.

At 10 am, it was 2^{\circ} \mathrm{F} in Fargo, North Dakota.

Adding and subtracting negative numbers FAQs

The additive inverse when you add any number to its opposite and the sum will always be zero. For example, 4+( - \; 4)=0.

Adding and subtracting negative numbers is similar to adding and subtracting whole numbers in a few ways. For one, you can represent both on a number line to support the visualization of adding and subtracting. In both, addition and subtraction are inverse operations of one another. Adding a negative value is equivalent to subtracting its absolute value, and subtracting a negative value is like adding its absolute value.

The next lessons are

• Multiplication and division
• Types of numbers
• Rounding numbers
• Adding and subtracting fractions
• Multiplying and dividing negative numbers

Still stuck?

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How to Add and Subtract Negatives

Last Updated: March 23, 2023 Fact Checked

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 72,251 times.

Problems with negative numbers may look difficult, but there's still only one right answer and with practice you can learn to find it quickly. There are at least two ways you can think your way through these problems. Most people start by learning on a number line.

Using a Number Line

• Here's another way to think of it: adding a negative number is the same as subtracting a positive number. 5 + (-2) = 5 - 2.

• Subtracting a negative number is the same as adding a positive number. 5 - (-2) = 5 + 2.

• Don't get confused by where you start on the number line. The first number only tells you where to begin on the number line. You'll always move right or left based on the type of problem and the second number.

• Here's a memory aid: it takes two lines to draw the two negative signs. It also takes two lines to draw a plus sign, so - - is the same as +, moving to the right.

Without a Number Line

• The absolute value of 6 is 6.
• The absolute value of -6 is also 6.
• 9 has a greater absolute value than 7.
• -8 has a greater absolute value than 5. It doesn't matter that one is negative.

• Rearrange it so you're subtracting the smaller absolute value from the larger one. Ignore the negative sign for now. For our example, write 4 - 2 instead.
• Solve that problem: 4 - 2 = 2. This isn't the answer yet!
• Look at the original problem and check the sign (+ or -) of the number with the largest absolute value number. 4 has a higher value than 2, so we look at that in the problem 2 + (-4). There's a negative sign in front of the 4, so our final answer will also have a negative sign. The answer is -2 .

• 3 - (-1) = 3 + 1 = 4
• (-2) - (-5) = (-2) + 5 = 5 - 2 = 3
• (-4) - (-3) = (-4) + 3 = 3 - 4 = -1

• (-7) - (-3) - 2 + 1
• = (-7) + 3 - 2 + 1
• = 3 - 7 - 2 + 1
• = (-4) - 2 + 1

Community Q&A

• The parentheses around negative numbers just make them easier to spot. You don't need to include them in our own work. Thanks Helpful 12 Not Helpful 0
• You can think of a negative number as debt, although this won't make sense for every problem. For instance, think of 40 + (-30) as having 40 dollars, and owing a debt of 30 dollars. After paying that debt, you end up with 40 + (-30) = 10. The same idea works if you have a debt of 40 dollars and get one more of 30 dollars: your total debt is -40 + (-30) = -70. Thanks Helpful 12 Not Helpful 5

You Might Also Like

• ↑ https://edu.gcfglobal.org/en/algebra-topics/negative-numbers/1/#
• ↑ https://www.chilimath.com/lessons/introductory-algebra/add-and-subtract-numbers-using-the-number-line/
• ↑ https://www.mathsisfun.com/positive-negative-integers.html
• ↑ https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negative-numbers-add-and-subtract/cc-7th-sub-neg-intro/a/subtracting-negative-numbers-review
• ↑ https://www.mathsisfun.com/numbers/absolute-value.html
• ↑ https://virtualnerd.com/middle-math/integers-coordinate-plane/add-integers/negative-number-addition
• ↑ https://edu.gcfglobal.org/en/algebra-topics/negative-numbers/1/

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Negative Numbers Worksheet Hub Page

Welcome to our Negative Numbers Worksheets hub page.

On this page, you will find links to all of our worksheets and resources about negative numbers.

Need help practicing adding, subtracting, multiplying or dividing negative numbers?

You've come to the right place!

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

• This page contains links to other Math webpages where you will find a range of activities and resources.
• If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

Negative Numbers Worksheet

We have a wide range of negative number resources to help you understand and use negative numbers.

Here are the following resources we currently have on our site:

• What are Negative Numbers

Negative Number Lines

• Comparing Negative Numbers

Adding Negative Numbers

Subtracting negative numbers.

• Mutiplying Negative Numbers

Dividing Negative Numbers

• Negative Number Games
• Absolute Value Worksheets

What are Negative Numbers?

Negative numbers are numbers with a value of less than zero.

They can be fractions, decimals, rational and irrational numbers.

-13, -½ , -√2, -6.4 and -123 are all negative numbers.

We have a page dedicated to learning about negative numbers below.

We have a selection of number lines, both filled and blank that have been designed to support learning and understanding with negative numbers.

One of our pages contains just negative number lines, the other page contains both positive and negative numbers.

• Number Lines with Negative and Positive Numbers
• Number Line Negative Numbers only

Back to Top

Comparing negative numbers

How to compare negative numbers

When you are comparing with negative numbers, everything swaps around and becomes a little more complicated!

With negative numbers, the more negative the number is, the lower its value.

As you go right along the number line, the values are increasing.

As you go left along the number line, the values are decreasing.

This means that any positive number (or even zero) will always be greater than any negative number.

• 0 > -3 this means 0 is greater than -3
• -8 < -5 this means -8 is less than -5
• -27 > -30 this means -27 is greater than -30
• -26 < 2 this means -26 is less than 2
• Ordering Negative Numbers -10 to 10

Randomly Generated Negative Number Worksheets

Our random worksheet generator will create a range of worksheets with values of your choice.

You can create your own unique worksheets complete with answers in seconds!

You can then choose to print or save your sheets for another time.

• Adding Positive and Negative Numbers (randomly generated)
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Adding & Subtracting Negative Numbers

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Multiplying Negative Numbers

• Negative Number Multiplication (randomly generated)
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Multiplying & Dividing Negative Numbers

• Multiply and Divide Negative Numbers (randomly generated)
• Negative Numbers Games

Take a look at our collection of negative numbers games.

We have a range of games of varying levels of difficulty.

Our games include:

• counting backwards along a number line (easiest)
• comparing and sequencing negative numbers
• subtracting with negative answers
• using all 4 operations to get a negative target number (hardest)

We have a selection of worksheets designed to help students learn about asbolute value.

Topics covered include:

• absolute value and opposite numbers
• comparing absolute values
• absolute value arithmetic
• solving absolute value equations

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Adding and Subtracting Negative Numbers

Intro Adding & Subtracting Multiplying & Dividing Exponents

What are the rules for adding and subtracting negative numbers?

The rules for adding and subtracting negative numbers works similarly to adding and subtracting positive numbers. When you'd added a positive number, you'd moved to the right on the number line. When you'd subtracted a positive number, you'd moved to the left.

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Adding and Subtracting Integers

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Now, if you're adding a negative number, you can regard this is pretty much the same as when you were subtracting a positive number, if you view "adding a negative" as adding to the left . That is, by plus-ing a minus, you're adding in the other direction. In the same vein, if you subtract a negative (that is, if you minus a minus), you're subtracting in the other direction; that is, you'll be subtracting by moving to the right .

Let's return to the first example from the previous page : 9 − 5 can also be written as 9 + (−5) . Graphically, it would be drawn as "an arrow from zero to nine, and then a 'negative' arrow going five units backwards":

← swipe to view full image →

...and you get 9 + (−5) = 4 .

Now look back at that subtraction you couldn't do : 5 − 9 . Because you now have negative numbers off to the left of zero, you also now have the room necessary in order to complete this subtraction. View the subtraction as adding a negative 9 ; that is, draw an arrow from zero to five, and then a "minus" arrow that goes nine units backwards:

...or, which is the same thing:

Then 5 − 9 = 5 + (−9) = −4.

Of course, this method of counting off your answer on a number line won't work so well if you're dealing with larger numbers. For instance, think about doing " 465 − 739 ". You certainly don't want to use a number line for this.

However, since 739 is larger than 465 , you know that the answer to 465 − 739 has to be a minus number, because "minus 739 " will take you somewhere to the left of zero. But how do you figure out which negative number is the answer?

Look again at 5 − 9 . You know now that the answer will be negative, because you're subtracting a number (that is, the 9 ) that is bigger than the one that you'd started with (that is, the 5 ). The easiest way of dealing with this is to do the subtraction normally (with the smaller number being subtracted from the larger number), and then put a "minus" sign on the answer:

9 − 5 = 4

5 − 9 = −4

This works the same way for bigger numbers (and is much simpler than trying to draw the picture). Since:

739 − 465 = 274

465 − 739 = −274

What is the rule for adding two negative numbers?

Adding two negative numbers is easy: you're just adding two "minus" arrows (that is, two arrows pointing to the left along the number line), so it's just like regular addition with positive numbers, but in the opposite direction. For instance, 4 +4 64 =4 10 , and −44 −4 6 =4 −44 +4 (−6) =4 −10 .

What is the rule for subtracting a negative number?

When you subtract a "minus" (that is, a negative) number from another number, you are actually adding a positive number to that other number. For instance, if you have −9 − (−3) , that second number represents two reversals. Thinking in terms of our arrows on our number line, we started with a −3 , being an arrow of length three and pointing to the left, and we subtracted it from the other number, thereby reversing the arrow's direction so that it's now an arrow of length three but pointing to the right.

Subtracting a negative number (that is, minus-ing a minus number) turns into its equivalent, which is adding a positive number (that is, plus-ing a plus number). In a sense, the two minuses "cancel" each other off; think of drawing a vertical line through both "minus" signs, turning them into "plus" signs:

1 − ( − 5) becomes 1 + ( + 5) = 6

(In practice, yes, you will be drawing an actual vertical line through the "minus" signs to create the "plus" signs.)

Why is "minus of a minus" equal to a plus?

This "minus of a minus is a plus" thing is actually a fairly important concept and, if you're asking why it works this way, then I'm assuming that your teacher's explanation didn't make much sense to you. So I won't give you a "proper" mathematical explanation of this "the minus of a minus is a plus" rule. Instead, here's a mental picture that I ran across some years ago:

Imagine that you're cooking some kind of stew in a big pot, but you're not cooking on a stove. Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number) to the pot, the temperature of the stew goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive.

Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, −2(−3) = +6 .

(The article by nrich uses a hot-air balloon with puffs of air and sandbags to accomplish the same thing.)

Here's another analogy that I've seen. Letting "good" be "positive" and "bad" be "negative", you could say:

good things happening to good people: a good thing

good things happening to bad people: a bad thing

bad things happening to good people: a bad thing

bad things happening to bad people: a good thing

To give a specific example of the above:

a family of four in their minivan gets home, safe and sound: a good thing

a drunk driver is veering all over the road in a stolen car: a bad thing

the family of four is killed by the drunk driver (he flees without a scratch): a bad thing

the drunk driver is caught and locked up before he hurts anybody: a good thing

Another model says:

When you're feeling down, listening to sad music can make you feel better.

Or think about driving a car:

Suppose "adding" means "driving in 'Drive'", "subtracting" means "driving in 'Reverse'", "positive" means "facing the right way for trafic on your side of the street", and "negative" means "facing the wrong way". To add a positive, you're driving with traffic. To subtract a positive, you're reversing into the car behind you. To add a negative, you're in "Drive" but, since you're facing the wrong way, you still plow into the car behind you. But if you subtract a negative, then you're facing the wrong way, but that's okay, because you're driving in "Reverse", so you're going with the flow of traffic. (Don't do this in real life! No cop is gonna buy the "but it's for my math class" excuse.)

The analogies above aren't technical explanations or proofs, but I hope they make the "minus of a minus is a plus" thing seem a bit more reasonable.

For some reason, it seems helpful to use the terms "plus" and "minus" instead, of "add, "subtract", "positive", and "negative". So, for instance, instead of saying "subtracting a negative", you'd say "minus-ing a minus". I have no idea why this is so helpful, but I do know that this verbal technique helped negatives click with me, too. (Thank you, Professor Mazumdar!)

(For an amusing and informative rant on the topic, try this " Math with Bad Drawings " article, which rejects "two negatives make a positive" in favor of "the opposite of the opposite of a thing is the thing itself".)

If somebody asks you, "Why does the minus of minus have to be a plus", you can (if you're wanting to be contrary) ask in reply, "Well, what *else* could it be?"

Suppose −(−1) = −1 instead of +1 . Then:

0 = 0(−1)

= (1 − 1)(−1)    ←(A)

= (1 + (−1))(−1)

= (1)(−1) + (−1)(−1)    ←(B)

= −1 + (−1)    ←(C)

But zero (where we started) does not equal 2 (where we ended up)! What happened?

I started with the number zero. At step (A) , I turned zero into 1 − 1 , which equals zero. At step (B) , I applied the Distributive Property. At step (C) , I applied the (wrong) assumption that the minus of a minus should be another minus (rather than a plus). The result of that assumption was a "proof" that zero equals something that is very much *not* zero. Since the only questionable part was step (C) , and since using (C) led to a contradiction (namely, saying that zero equals two), then (C) must be wrong. Ergo, minus of a minus has to be a plus.

(This is, by the way, an example of a "proof by contradiction", wherein you assume something that is the opposite of what is true [in this case, assuming that the minus of a minus is another minus], show that this assumption leads to a contradiction of known good information [namely, that zero equals two], and thereby prove that the assumption was false, so the opposite is true [namely, that the minus of a minus is a plus].)

A note to native-English speakers: Don't use the "a double negative is a positive" thing, because that's a thing pretty much only in ("proper") English.

We've seen how to work with two numbers. What if we're adding and subtracting lots of numbers? The main thing to be careful of is that signs stay with their numbers.

• Simplify 18 − (−16) − 3 − (−5) + 2

Probably the simplest thing to do is convert everything to addition, group the positives together and the negatives together, combine, and simplify. It looks like this:

18 − (−16) − 3 − (−5) + 2

= 18 + 16 − 3 + 5 + 2

= 18 + 16 + (−3) + 5 + 2

= 18 + 16 + 5 + 2 + (−3)

= 41 + (−3)

= 41 − 3

In my working above, I used the fact that the minus of a minus is a plus to convert −(−16) to +16 and −(−5) to +5 . To help me keep the signs with their numbers, I also converted −3 to +(−3) . That way, when I moved the −3 , I would be sure that the "minus" sign moved with it.

• Simplify −43 − (−19) − 21 + 25

−43 − (−19) − 21 + 25

= −43 + 19 − 21 + 25

= (−43) + 19 + (−21) + 25     *

= (−43) + (−21) + 19 + 25     *

= (−64) + 44

= 44 + (−64)

= 44 − 64

= −20

I got that last value by noting that 64 − 44 = 20 , that 64 is larger than 44 , and that 64 has the "minus" sign. so my answer is:

To get from the first starred line above to the second, I moved terms around to get all the plusses together and all the minusses together. It's okay for me to move the terms around, as long as I move their signs around *with* their numbers. When you're just starting out, you may find it helpful to convert "minus [some number]" to "plus (the minus of [that number])", which can make it easier to keep track of where those "minus" signs need to go.

• Simplify 84 + (−99) + 44 − (−18) − 43

I'll start by converting the minus of minusses to plusses, and the minusses to plusses of minusses. This will help me keep the signs straight:

84 + (−99) + 44 − (−18) − 43

= 84 + (−99) + 44 + 18 + (−43)

= 84 + 44 + 18 + (−99) + (−43)

= 146 + (−142)

= 146 − 142

What are the rules for adding and subtracting with negative numbers?

The take-aways from this page are the following rules for adding and subtracting with negative numbers:

• If you're adding two negative numbers, then add in the usual way, remembering to put a "minus" sign on the result. Example: −2 + (−3) = −5
• If you're adding a positive number and a negative number, subtract the smaller number (that is, the number that's closer to zero) from the larger number (that is, the number that's further from zero), and use the sign for whatever was the larger number on your answer. Example: +2 + −3 = −(3 − 2) = −(1) = −1
• If you're subtracting a negative number, then cancel the "minus" signs to convert to adding a positive. Example: −2 − (−3) = −2 + 3 = +1

As long as you're careful about moving signs *with* their numbers, you should be okay.

URL: https://www.purplemath.com/modules/negative2.htm

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These negative numbers worksheets will have your kids working with positive and negative integers in no time! Starting with adding and subtracting negative numbers, and gradully work up to multiplying and dividing negative numbers, multiplying multi-digit negative numbers and long division for negative numbers.

Adding and Subtracting Negative Numbers

36 negative numbers worksheets.

Worksheets for adding negative numbers and subtracting negative numbers.

Multiplying and Dividing Small Negative Numbers

16 negative numbers worksheets.

The worksheets in this section introduce negative numbers integers in multiplication and division math problems. All problems deal with smaller integers that can be solved without multi-digit multiplication or long division.

Multiple-Digit Multiplication with Negative Numbers

If you have mastered basic multiplication with negative integers, these worksheets for multiple digit multiplication will give your negative number skills a more thorough test.

Long Division with Negative Numbers

Ready to keep your signs straight? These long division worksheets have negative divisors and negative quotients (or both!). Some negative division problems include remainders.

Solving Math Problems with Negative Numbers

Negative numbers is a math topic that typically comes into play around 6th grade, and it's introduced as part of the Common Core standard at that grade level.

Negative numbers appear in a variety of situations in applied math. Often you'll see negative numbers directly in measurements, for example measuring altitude above or below sea level, temperature above or below freezing or in financial applications with positive and negative amounts of money. A more frequent, but also more abstract, application of negative numbers is dealing with rates of change. You will also encounter negative values in geometry when graphing in various quadrants on a coordinate plane. And of course, as you make your way into algebra and more advanced geometry, negative numbers play an increasingly important role.

Kids in the late primary grades should be capable of reasoning about negative integers on the number line, and this is usually a good place to start exploring the basic math operations with negative numbers. This is also a good way to start visualizing how the rules for signed numbers work. The two critical ones to learn are that a subtracting a negative number is the same as addition, and that multiplying two negative numbers yields a positive product. Most of the other behaviors of negative numbers with the conventional math operations seem to be straightforward and intuitive, but memorizing those two rules will give your grade schoolers a solid start. For more on the rules for managing signs with negative numbers for the various operations, see the respective worksheet pages for a complete discussion and tips.

The worksheets on this page introduce adding and subtracting negative numbers, as well as multiplying and dividing negative numbers. The initial sets deal with small integers before moving on to multi-digit multiplication and long division with negatives. Regardless of where you're at in your process of learning negative numbers, these worksheets will give your students plenty of practice when they need to master this often negative topic!

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Part 2: Addition and Subtraction

2.4: Word problems for adding and subtracting with negative integers

Real-world interpretations of negative integers.

We can work with negative integers more easily if we can interpret them using meaningful real-world phenomena, such as temperatures below zero, building floors below ground, or amounts of money owed. For example, –4 could represent a fridge temperature of –4ºC, a parking level 4 floors below ground, or a $4 debt. Some other ideas for negative amounts that lend themselves to story problems are:

• Hot-air balloons with puffs of gas (positive) and sand bags (negative). Adding a puff of gas or taking away a sand bag makes the balloon go up by 1 metre, while taking away a puff of gas or adding a sand bag makes the balloon go down by 1 metre.
• Magic soup with hot cubes (positive) and cold cubes (negative). Adding a hot cube or taking away a cold cube makes the soup temperature go up by 1 degree, while taking away a hot cube or adding a cold cube makes the soup temperature go down by 1 degree.
• People with happy thoughts (positive) and sad thoughts (negative). Adding a happy thought or taking away a sad thought makes the person’s happiness go up by 1 level, while taking away a happy thought or adding a sad thought makes the person’s happiness go down by 1 level.

Adding a negative integer to a positive integer

Add To Problem : Add a negative amount of an object to a positive amount of that same object. For example, add a $4 debt to a $7 credit. Recall from 3b: Adding and subtracting with negative integers that we use the subtraction algorithm to calculate A + (–B) by rewriting it as A – B (if A > B) or –(B – A) (if A < B).

Here’s an example of an “add to” problem involving a negative integer:

• Arithmetic problem: Solve 7 + (–4).
• Word problem: Keenan has $7 in his pocket but he owes his friend $4. How much money does Keenan have to spend?
• 7 + (–4) = 3, so Keenan has $3 to spend.

Subtracting a negative integer from a positive integer

Take From Problem : Taking away something negative has the same effect as adding something positive. For example, taking away a $4 debt has the same effect as adding a $4 credit. Recall from  3b: Adding and subtracting with negative integers that we use the addition algorithm to calculate A – (–B) by rewriting it as A + B.

Here’s an example of a “take from” problem involving a negative integer:

• Arithmetic problem: Solve 7 – (–4).
• Word problem: A hot-air balloon with 4 puffs of gas and 4 sand bags is balanced (neither going up nor going down). What happens to the balloon if we add 7 more puffs of gas and take away the 4 sand bags?
• 7 – (–4) = 11, so the ballon goes up by 11 metres.

Compare Problem : By comparing a positive integer with a negative integer, we can see that the difference between A and –B is the same as the sum A + B. For example:

• Word problem: At 12 noon it is 7ºC in Vancouver and –4ºC in Whistler. How much warmer is it in Vancouver than in Whistler?
• 7 – (–4) = 11, so it is 11ºC warmer in Vancouver than in Whistler.

The video below works through some examples of word problems for adding and subtracting with negative integers.

Practice Exercises

Do the following exercises to practice matching negative integer addition/subtraction problems and word problems.

Mathematics For Elementary Teachers Copyright © 2023 by Iain Pardoe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Adding and Subtracting Positive and Negative Numbers

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We can subtract a negative number from a negative number and end up with a negative solution:

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Adding Negative Numbers

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